{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:CWL7QU36TRON55RT7IZK6CSXHI","short_pith_number":"pith:CWL7QU36","canonical_record":{"source":{"id":"1908.01037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-02T20:12:38Z","cross_cats_sorted":[],"title_canon_sha256":"efe74df87c3bb32600a09cb506e17b0edd672e10ca33835ef64322910110a13d","abstract_canon_sha256":"914c3013d041c2ee9df258cd7b42750eac07a88b428bc314c0bc177d3bb6a704"},"schema_version":"1.0"},"canonical_sha256":"1597f8537e9c5cdef633fa32af0a573a2b40f61d692f318d7e626b2056ace2f4","source":{"kind":"arxiv","id":"1908.01037","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01037","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01037v1","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01037","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_12","alias_value":"CWL7QU36TRON","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_16","alias_value":"CWL7QU36TRON55RT","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_8","alias_value":"CWL7QU36","created_at":"2026-07-04T23:51:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:CWL7QU36TRON55RT7IZK6CSXHI","target":"record","payload":{"canonical_record":{"source":{"id":"1908.01037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-02T20:12:38Z","cross_cats_sorted":[],"title_canon_sha256":"efe74df87c3bb32600a09cb506e17b0edd672e10ca33835ef64322910110a13d","abstract_canon_sha256":"914c3013d041c2ee9df258cd7b42750eac07a88b428bc314c0bc177d3bb6a704"},"schema_version":"1.0"},"canonical_sha256":"1597f8537e9c5cdef633fa32af0a573a2b40f61d692f318d7e626b2056ace2f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:51:25.803145Z","signature_b64":"t+MS60OzI+tkshltV9noqQiWNBVbC6+QW6kRQvACQ/IW5d2R67OpXIrWyXIdFg3o7/IJrReSJMP4XlSbOQODDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1597f8537e9c5cdef633fa32af0a573a2b40f61d692f318d7e626b2056ace2f4","last_reissued_at":"2026-07-04T23:51:25.802797Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:51:25.802797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.01037","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:51:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M4+4gtR3v+Pmv+9XEFkQmxdTpzJGddt+HRm9njqefmJKFre+EmHQkXOoxdD20pBa5igDCERpixaVBZpgNBSLAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T10:00:15.335963Z"},"content_sha256":"beac90a426be22b116090732d1fb2084b1afd968013321bcc1c0ae363e124b33","schema_version":"1.0","event_id":"sha256:beac90a426be22b116090732d1fb2084b1afd968013321bcc1c0ae363e124b33"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:CWL7QU36TRON55RT7IZK6CSXHI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximating Pointwise Products of Quasimodes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mei Ling Jin","submitted_at":"2019-08-02T20:12:38Z","abstract_excerpt":"We obtain approximation bounds for products of quasimodes for the Laplace-Beltrami operator, on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes $uv$ by a low-degree vector space $B_{n}$, and we prove that the size of the space $\\dim(B_{n})$ is small. In our paper, we first study bilinear quasimode estimates of all dimensions $d = 2, 3$, $d = 4,5$ and $d \\ge 6$, respectively, to make the highest frequency disappear from the right hand. Furthermore, the result of the case $\\lambda=\\mu$ of bilinear quasimode estimates improves $L^{4}$ qua"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01037","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01037/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:51:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yCcOPpPdZ7Ng/3Qit8dPlm1j4lsttEoxYuZarATQFB+Gi+pM2tjAMG/V6eo3KumLJbGxHIWgZkefF4R3HneMAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T10:00:15.337584Z"},"content_sha256":"b3c61f02bec00fddb5cdd33968d1166421463195c29eae2249b7f15932784aa2","schema_version":"1.0","event_id":"sha256:b3c61f02bec00fddb5cdd33968d1166421463195c29eae2249b7f15932784aa2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CWL7QU36TRON55RT7IZK6CSXHI/bundle.json","state_url":"https://pith.science/pith/CWL7QU36TRON55RT7IZK6CSXHI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CWL7QU36TRON55RT7IZK6CSXHI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T10:00:15Z","links":{"resolver":"https://pith.science/pith/CWL7QU36TRON55RT7IZK6CSXHI","bundle":"https://pith.science/pith/CWL7QU36TRON55RT7IZK6CSXHI/bundle.json","state":"https://pith.science/pith/CWL7QU36TRON55RT7IZK6CSXHI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CWL7QU36TRON55RT7IZK6CSXHI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:CWL7QU36TRON55RT7IZK6CSXHI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"914c3013d041c2ee9df258cd7b42750eac07a88b428bc314c0bc177d3bb6a704","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-02T20:12:38Z","title_canon_sha256":"efe74df87c3bb32600a09cb506e17b0edd672e10ca33835ef64322910110a13d"},"schema_version":"1.0","source":{"id":"1908.01037","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01037","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01037v1","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01037","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_12","alias_value":"CWL7QU36TRON","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_16","alias_value":"CWL7QU36TRON55RT","created_at":"2026-07-04T23:51:25Z"},{"alias_kind":"pith_short_8","alias_value":"CWL7QU36","created_at":"2026-07-04T23:51:25Z"}],"graph_snapshots":[{"event_id":"sha256:b3c61f02bec00fddb5cdd33968d1166421463195c29eae2249b7f15932784aa2","target":"graph","created_at":"2026-07-04T23:51:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.01037/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We obtain approximation bounds for products of quasimodes for the Laplace-Beltrami operator, on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes $uv$ by a low-degree vector space $B_{n}$, and we prove that the size of the space $\\dim(B_{n})$ is small. In our paper, we first study bilinear quasimode estimates of all dimensions $d = 2, 3$, $d = 4,5$ and $d \\ge 6$, respectively, to make the highest frequency disappear from the right hand. Furthermore, the result of the case $\\lambda=\\mu$ of bilinear quasimode estimates improves $L^{4}$ qua","authors_text":"Mei Ling Jin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-02T20:12:38Z","title":"Approximating Pointwise Products of Quasimodes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01037","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:beac90a426be22b116090732d1fb2084b1afd968013321bcc1c0ae363e124b33","target":"record","created_at":"2026-07-04T23:51:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"914c3013d041c2ee9df258cd7b42750eac07a88b428bc314c0bc177d3bb6a704","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-02T20:12:38Z","title_canon_sha256":"efe74df87c3bb32600a09cb506e17b0edd672e10ca33835ef64322910110a13d"},"schema_version":"1.0","source":{"id":"1908.01037","kind":"arxiv","version":1}},"canonical_sha256":"1597f8537e9c5cdef633fa32af0a573a2b40f61d692f318d7e626b2056ace2f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1597f8537e9c5cdef633fa32af0a573a2b40f61d692f318d7e626b2056ace2f4","first_computed_at":"2026-07-04T23:51:25.802797Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:51:25.802797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t+MS60OzI+tkshltV9noqQiWNBVbC6+QW6kRQvACQ/IW5d2R67OpXIrWyXIdFg3o7/IJrReSJMP4XlSbOQODDw==","signature_status":"signed_v1","signed_at":"2026-07-04T23:51:25.803145Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.01037","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:beac90a426be22b116090732d1fb2084b1afd968013321bcc1c0ae363e124b33","sha256:b3c61f02bec00fddb5cdd33968d1166421463195c29eae2249b7f15932784aa2"],"state_sha256":"d978dc115a9f47cec965d2d8839307f77842803cd7a4e02751f7b942d3efe272"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DTkt4bbC3/HvLyS6tOa7BkDk3d1dJK03ydvU0LV6FJdNT6a8Wk+HikMsl2ypeplutrZie8iLHaLc5KCNdnTJCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T10:00:15.346062Z","bundle_sha256":"db4eab5097e5028c9be3127cafdc5c8e85dc8421e1479a81aba384ee572a38a0"}}